Finding porosity and oil fraction by means of dielectric spectroscopy

ABSTRACT

Methods, systems, devices and products for evaluating an earth formation. Methods include estimating at least one property of the earth formation using at least one polarization parameter estimated using a real part and an imaginary part of a permittivity of the earth formation at a plurality of frequencies, where the real parts and the imaginary parts are based on measurements obtained using an electromagnetic tool in a borehole penetrating the earth formation.

BACKGROUND OF THE DISCLOSURE

1. Field of the Disclosure

This disclosure generally relates to exploration and production ofhydrocarbons involving investigations of regions of an earth formationpenetrated by a borehole. More specifically, the disclosure relates tothe dielectric spectroscopy of an earth formation using a logging toolin a borehole.

2. Description of the Related Art

Electrical earth borehole logging is well known to persons having anordinary level of skill in the art, and various devices and varioustechniques have been described for this purpose. Broadly speaking, thereare two categories of electrical logging apparatus. In the firstcategory, one or more measurement electrodes—current source(s) orsink(s)—are used in conjunction with a return electrode (which may be adiffuse electrode such as a logging tool's body or mandrel). Ameasurement current flows in a circuit that connects a current source tothe measurement electrode(s), through the earth formation to the returnelectrode, and back to the current source in the tool. In a secondcategory, that of inductive measuring tools, an antenna within themeasuring instrument induces a current flow within the earth formation.The magnitude of the induced current is detected using either the sameantenna or a separate receiver antenna. The present disclosure belongsto the second category.

SUMMARY OF THE DISCLOSURE

In view of the foregoing, the present disclosure is directed to a methodand apparatus for estimating at least one property using dielectricspectroscopy of subterranean formations penetrated by a borehole.

One embodiment according to the present disclosure includes a method ofevaluating an earth formation including estimating at least one propertyof the earth formation using at least one polarization parameterestimated using a real part and an imaginary part of a permittivity ofthe earth formation at a plurality of frequencies, where the real partsand the imaginary parts are based on measurements obtained using anelectromagnetic tool in a borehole penetrating the earth formation.

The at least one property may include at least one of: i) porosity andii) oil fraction. The earth formation may be or include a saturatedporous medium. The saturation may be substantially due to water and oil.The method may also include conveying the electromagnetic tool in theborehole. The electromagnetic tool may use electrical induction. Themethod may also include using the electromagnetic tool for making themeasurements at the plurality of frequencies. Using the real part andthe imaginary part may include generating a spectral dielectric curvebased on the real parts and imaginary parts; estimating a high-frequencylimit for the real parts using the spectral dielectric curve; andestimating an angle between the spectral dielectric curve and an axisformed at the high-frequency limit. Using the real part and theimaginary part may include modeling the permittivity as a relaxationcurve. Using the real part and the imaginary part may include modelingthe permittivity as a Havriliak-Negami curve. Using the real part andthe imaginary part may include estimating a plurality of spectralcharacteristics of the Havriliak-Negami curve. The polarizationparameter may be an oil saturation parameter. The polarization parametermay be a water saturation parameter.

One embodiment according to the present disclosure includes an apparatusfor evaluating an earth formation. The apparatus may include a carrierconfigured to be conveyed in a borehole penetrating the earth formation;a electromagnetic tool disposed on the carrier and configured to makemeasurements indicative of an imaginary part and a real part of apermittivity of the earth formation at a plurality of frequencies; andat least one processor. The processor may be configured to: estimate thereal part and the imaginary part of the permittivity of the earthformation at the plurality of frequencies; estimate at least onepolarization parameter using the using the real parts and imaginaryparts; and estimate at least one property of the earth formation usingthe at least one polarization parameter.

The at least one property may include at least one of: i) porosity andii) oil fraction. The earth formation may be or include a saturatedporous medium. The saturation may be substantially due to water and oil.The electromagnetic tool may use electrical induction. The processor maybe configured to use the electromagnetic tool for making themeasurements at the plurality of frequencies. Using the real part andthe imaginary part may include generating a spectral dielectric curvebased on the real parts and imaginary parts; estimating a high-frequencylimit for the real parts using the spectral dielectric curve; andestimating an angle between the spectral dielectric curve and an axisformed at the high-frequency limit. Using the real part and theimaginary part may include modeling the permittivity as a relaxationcurve. Using the real part and the imaginary part may include modelingthe permittivity as a Havriliak-Negami curve. Using the real part andthe imaginary part may include estimating a plurality of spectralcharacteristics of the Havriliak-Negami curve. The polarizationparameter may be an oil saturation parameter. The polarization parametermay be a water saturation parameter.

One embodiment includes a non-transitory computer-readable mediumproduct having instructions thereon that, when executed, cause at leastone processor to perform a method including estimating at least oneproperty of the earth formation using at least one polarizationparameter estimated using a real part and an imaginary part of apermittivity of the earth formation at a plurality of frequencies, wherethe real parts and the imaginary parts are based on measurementsobtained using an electromagnetic tool in a borehole penetrating theearth formation. The non-transitory computer-readable medium product mayinclude at least one of: (i) a ROM, (ii) an EPROM, (iii) an EEPROM, (iv)a flash memory, or (v) an optical disk.

The at least one property may include at least one of: i) porosity andii) oil fraction. The earth formation may be or include a saturatedporous medium. The saturation may be substantially due to water and oil.The method may also include conveying the electromagnetic tool in theborehole. The electromagnetic tool may use electrical induction. Themethod may also include using the electromagnetic tool for making themeasurements at the plurality of frequencies. Using the real part andthe imaginary part may include generating a spectral dielectric curvebased on the real parts and imaginary parts; estimating a high-frequencylimit for the real parts using the spectral dielectric curve; andestimating an angle between the spectral dielectric curve and an axisformed at the high-frequency limit. Using the real part and theimaginary part may include modeling the permittivity as a relaxationcurve. Using the real part and the imaginary part may include modelingthe permittivity as a Havriliak-Negami curve. Using the real part andthe imaginary part may include estimating a plurality of spectralcharacteristics of the Havriliak-Negami curve. The polarizationparameter may be an oil saturation parameter. The polarization parametermay be a water saturation parameter.

BRIEF DESCRIPTION OF THE DRAWINGS

The present disclosure is best understood with reference to theaccompanying figures in which like numerals refer to like elements andin which:

FIG. 1 is a schematic of a drilling site including an electromagnetictool for estimating a parameter in an earth formation according to oneembodiment of the present disclosure;

FIG. 2A is a schematic view of an electromagnetic tool in accordancewith one embodiment of the present disclosure;

FIG. 2B is a schematic view of a pad of an electromagnetic tool inaccordance with one embodiment of the present disclosure;

FIGS. 3A and 3B are schematics of antenna configurations for a pad of anelectromagnetic tool for embodiments according to the presentdisclosure;

FIG. 4A is a schematic of an antenna configuration for a pad of anelectromagnetic tool for another embodiment according to the presentdisclosure;

FIG. 4B is a schematic of an antenna configuration for a pad of anelectromagnetic tool for another embodiment according to the presentdisclosure;

FIG. 5 is a flow chart for a method for one embodiment according to thepresent disclosure;

FIG. 6 is a set of graphs showing relaxations for different polarizationtypes for one embodiment according to the present disclosure;

FIG. 7 is a graph of complex permittivity for water and water/oilsaturated sandstone for one embodiment according to the presentdisclosure;

FIG. 8 is a graph of imaginary versus real parts of complex permittivityfor water and water/oil saturated sandstone for one embodiment accordingto the present disclosure;

FIG. 9A shows the graph of α=α(β) for two values of ν;

FIG. 9B shows the graph of α_(*)=α_(*)(β_(*)); and

FIG. 10 is a flow chart for a method for one embodiment according to thepresent disclosure.

DETAILED DESCRIPTION OF THE DISCLOSURE

In the disclosure that follows, in the interest of clarity, not allfeatures of actual implementations are described. It will of course beappreciated that in the development of any such actual implementation,as in any such project, numerous engineering and technical decisionsmust be made to achieve the developers' specific goals and subgoals(e.g., compliance with system and technical constraints), which willvary from one implementation to another. Moreover, attention willnecessarily be paid to proper engineering and programming practices forthe environment in question. It will be appreciated that suchdevelopment efforts may be complex and time-consuming, outside theknowledge base of typical laymen, but would nevertheless be a routineundertaking for those of ordinary skill in the relevant fields.

Dielectric spectroscopy includes the estimation of electric permittivityof materials. Electric permittivity may include imaginary and realparts, which may vary with the frequency of an electrical signal exposedto a material. In some aspects, electric permittivity may be estimatedusing an electromagnetic tool configured to generated an electriccurrent at a plurality of frequencies.

In some aspects, the electromagnetic tool may include an inductiveelectromagnetic probe that may be electrically coupled to an earthformation from inside of a borehole penetrating the earth formation.Dielectric permittivity may be obtained by solving Maxwell's equations.For a homogeneous medium, the relationship between a magnetic field,H_(z), and dielectric permittivity, ∈, may be expressed as:

$\begin{matrix}{{H_{r}^{0} = {\frac{M_{z}{rz}}{4\;\pi\; R^{5}}\left( {3 + {3\;{kR}} + {k^{2}R^{2}}} \right)e^{- {kR}}}},{H_{\varphi}^{0} = 0},{H_{z}^{0} = {{- \frac{M_{z}}{4\;\pi\; R^{3}}}\left( {\frac{3\; r^{2}}{R^{2}} + \frac{3\;{kr}^{2}}{R} + {k^{2}R^{2}} - 2 - {2\;{kR}}} \right)e^{- {kR}}}},{R^{2} = {r^{2} + z^{2}}},{k^{2} = {{- \omega^{2}}\mu\; ɛ}},{ɛ = {ɛ^{\prime} + {i\;{ɛ^{''}.}}}}} & (1)\end{matrix}$where M_(Z) is the receiver coil magnetic moment, k is a wave number, ωis a circular ( ) frequency, R is a radial distance, r and z arecoordinates in the cylindrical coordinate system, μ is the permeabilityof the material, and ∈′ and ∈″ are real and imaginary parts ofdielectric permittivity. The dielectric permittivity may be estimatedusing methods known to those of skill in the art, including, but notlimited to one or more of: (i) a Newton method and (ii) aMarquardt-Levenberg method.

The estimated imaginary and real parts of permittivity of the earthformation may be used to estimate at least one parameter of interest ofthe earth formation. The at least one parameter of interest may include,but is not limited to: (i) water saturation, (ii) oil saturation, and(iii) percentage of oil; (iv) percentage of water; and (v) formationporosity.

If, for example, a dipole antenna configured to emit electromagneticwaves is placed in the center of the borehole with a receiver antennalocated in the borehole at a certain distance from the transmitter, thenreal and imaginary parts of the magnetic field may be measured by thereceiver antenna. A spectral image of dielectric permittivity may begenerated using the imaginary and real parts of permittivity over aplurality of frequencies. The spectral image may also be referred to asa “spectral dielectric curve” and a “polarization curve”. Each point onthe spectral image of dielectric permittivity may correspond to aspecific frequency in the electromagnetic spectrum. The bulk fractionsof water and oil in the earth formation may be estimated using thespectral image of dielectric permittivity.

Aspects of the present disclosure include using a borehole inductiveelectromagnetic tool to measure the real and imaginary parts of thedielectric spectrum at a fixed borehole depth. This may includeestimating all spectral characteristics of the Havriliak-Negami curve.The percentage of oil contained in the formation pores may be estimatedfrom oil and/or water saturation characteristics using methods disclosedherein. Using the tabulated curves for sandstones, carbonates, etc. andthe limit value of another polarization parameter, further aspects ofthe disclosure may enable estimation of the formation porosity. Exampleapparatus and method embodiments for estimating properties of theformation are discussed below.

FIG. 1 shows an electromagnetic tool 10 suspended, in a borehole 12penetrating earth formation 13, from a suitable cable 14 that passesover a sheave 16 mounted on drilling rig 18. By industry standard, thecable 14 includes a stress member and seven conductors for transmittingcommands to the tool and for receiving data back from the tool as wellas power for the tool. The electromagnetic tool 10 is raised and loweredby draw works 20. Electronic module 22, on the surface 23, transmits therequired operating commands downhole and in return, receives data backwhich may be recorded on an archival storage medium of any desired typefor concurrent or later processing. The data may be transmitted inanalog or digital form. Data processors such as a suitable computer 24,may be provided for performing data analysis in the field in real timeor the recorded data may be sent to a processing center or both for postprocessing of the data. Some or all of the processing may also be doneby using a downhole processor at a suitable location on the logging tool10. While a wireline conveyance system has been shown, it should beunderstood that embodiments of the present disclosure may be utilized inconnection with tools conveyed via rigid carriers (e.g., jointed tubularor coiled tubing) as well as non-rigid carriers (e.g., wireline,slickline, e-line, etc.). Some embodiments of the present disclosure maybe deployed along with LWD/MWD tools.

The electromagnetic tool 10 may include at least one transmittingantenna and at least two receiving loop antennas mounted on a pad. Thetool may be operated in at least two modes. A first mode may be referredto as Mini-MPR (Multiple propagation resistivity) mode that may measureattenuation and a phase difference between the two receivers. Theelectromagnetic tool 10 may also be operated in a second mode (aninduction mode) in which a compensated magnetic field (voltage) may bemeasured. The current in the transmitter coil may induce a magneticfield in the earth formation 13. This magnetic field, in turn, may causeeddy currents to flow in the earth formation 13. Because of the presenceof these formation currents, a magnetic field may be coupled into areceiver coil R thereby generating a receiver signal. Logging toolshaving “a receiver coil” and “a transmitter coil” each comprised ofseveral coils arranged in a predetermined fashion to obtain a desiredresponse may be used. The receiver signal may then be amplified andapplied to one or more phase sensitive detectors (PSDs). Each PSD maydetect a phase component signal having a phase identical to a phasereference signal which may also be applied to the detector. The phasereference signal may have a predetermined phase relationship to thecurrent in the transmitter coil(s). The output of the PSD(s) may befurther processed downhole, or may be sent uphole to surface equipmentfor processing or display to an operating engineer.

In the induction mode, one receiver loop coil may serve as a mainreceiver and the other as a bucking receiver. The transmitting antennasmay include loops and/or electric dipoles. For loop transmitterantennas, the transmitters and receivers may be in one of threeorientations. If the z-axis of the tool is parallel to the longitudinalaxis of the tool, then the x-axis may be radial through the center ofthe pad, and the y-axis may be tangential to the pad. The zz-componentmay refer to a z-source and a z-receiver and so on. In some embodiments,xx-transmitters and receivers may be used.

FIG. 2A shows an electromagnetic tool 10 for one embodiment according tothe present disclosure. The electromagnetic tool 10 may include a body55 with two pads 51A, 51B extended on extension devices 53A, 53B. Twopads are shown for illustrative purposes and, in actual practice, theremay be more pads. The extension devices 53A, 53B may be electricallyoperated, electromechanically operated, mechanically operated orhydraulically operated. With the extension devices 53A, 53B fullyextended, the pads 51A, 51B can make contact with the borehole wall (notshown) and make measurements indicative of properties of the boreholewall. Orientation sensors (not shown) may provide an indication of theorientation of the electromagnetic tool 10. In addition, cable depthmeasurements may be obtained using a sensor (not shown) at the surfacethat measures the amount of cable spooled out. In addition,accelerometers may be used downhole to provide other measurementsindicative of the depth of the electromagnetic tool 10. The orientationsensors may include accelerometers, magnetometers or gyroscopes. Depthmay also be estimated from a gyro output.

An exemplary arrangement of dual transmitters and receivers on each ofthe pads is shown in FIG. 2B. Shown therein is pad 51A with twotransmitters 55A, 55B disposed about two receivers 57A, 57B. Alsodepicted schematically by arrows in FIG. 2B are measurements that may bemade by each of the two receivers 57A, 57B corresponding to signalsgenerated by each of the two transmitters 55A, 55B.

The use of dual transmitters may provide a symmetrical response. The useof dual transmitters may also reduce effects of borehole rugosity. Also,the use of dual transmitters may reduce electronics-related errors inattenuation measurement. The electronics-related errors may not affectthe phase difference measurement.

When in the Mini-MPR mode, the two transmitters 55A, 55B may be placedsymmetrically with respect to the receiver antennas 57A, 57B.Attenuation and phase difference are measured for each of thetransmitters 55A, 55B. The measurements may be averaged to give thefinal readings:

$\begin{matrix}{{{{Att} = \frac{{Att}_{T\; 1} + {Att}_{T\; 2}}{2}};}{{Pha} = \frac{{Pha}_{T\; 1} + {Pha}_{T\; 2}}{2}}} & (2)\end{matrix}$where the subscripts T1 and T2 denote the first and second receivers.Assuming a uniform earth formation for which the magnetic fields at thereceiver locations are H₁ and H₂ and assuming that the two receivershave gains G₁ and G₂, the ratio of the two receiver outputs R_(T1) forthe 1st transmitter may be derived from the ratio:

$\begin{matrix}{R_{T\; 1} = {\frac{G_{2}H_{2}}{G_{1}H_{1}} = {\frac{G_{2}}{G_{1}}\frac{A_{2}}{A_{1}}e^{i\;\Delta\;\phi}}}} & (3)\end{matrix}$where A₁ and A₂ are the amplitudes of H₁ and H₂, respectively; Δφ is thephase difference between the two receivers. From eqn. (3) it follows

$\begin{matrix}{{{Att}_{T\; 1} = {{{- 20}\;\log\frac{G_{2}}{G_{1}}} - {20\;\log\frac{A_{2}}{A_{1}}}}},} & (4) \\{{Pha}_{T\; 1} = {\Delta\;{\phi.}}} & (5)\end{matrix}$Thus, it is clear that the gain change affects attenuation measurementbut not the phase difference measurement.

Similarly, attenuation measurement for the 2nd transmitter is derivedfrom

$\begin{matrix}{R_{T\; 1} = {\frac{G_{1}H_{2}}{G_{2}H_{1}} = {\frac{G_{1}}{G_{2}}\frac{A_{2}}{A_{1}}{e^{i\;\Delta\;\phi}.}}}} & (6)\end{matrix}$

The attenuation measurement may be written as:

$\begin{matrix}{{Att}_{T\; 1} = {{{- 20}\;\log\frac{G_{1}}{G_{2}}} - {20\;\log{\frac{A_{2}}{A_{1}}.}}}} & (7)\end{matrix}$

Averaging equations (3) and (4) may remove the effect of gain variation.Those versed in the art would recognize that measurements of amplitudeand phase can, in addition to resistivity determination, also be usedfor determining the dielectric constant of the earth formation.

FIG. 3A shows a schematic of a generic tool configuration for oneembodiment according to the present disclosure. Here, multiple receiverpairs of receivers may be used to achieve sufficient azimuthal coverage.Pad 51A may include two receiver arrays 103A, 103B. For each receiver105 in the upper array 103A, there is a corresponding receiver 105 inthe lower array 103B. In one embodiment, the coils 101A, 101B of theupper and lower receiver arrays may be aligned radially with respect tothe tool axis (movement) direction. The receiver coils 105 are separatedlaterally by a constant distance that is determined by the azimuthalresolution of the electromagnetic tool. Two transmitting antennas 101A,101B may be placed above receiver array 103A and below receiver array103B. The transmitting antennas 101A, 101B may be operated one at a timeduring which measurements from each and every receiver pair are made. Anexemplary current flow direction for the transmitters 101A, 101B isshown by the arrows in FIG. 3A. With the indicated current flow of thetransmitters 101A, 101B and the coil orientation of the receivers 105,the measurements made would be xx-measurements. The measurements mayinclude attenuation rate, phase difference, or compensated magneticfield.

FIG. 3B shows a schematic of another embodiment of a generic toolconfiguration with staggered receiver pairs according to the presentdisclosure. Depending on the size of the receiver coils 105, thereceiver pairs may be staggered in the tool axis direction, allowing asmall separation between the receiver pairs. The upper receiver array103A′ may comprise two staggered rows of receivers 105 and the lowerreceiver array 103B′ may comprise two staggered rows of receivers 105 toreduce the gaps in azimuthal coverage of the configuration of FIG. 3A.

FIG. 4A is a schematic of a transmitter for one embodiment according tothe present disclosure. Transmitters 101A′ and 101B′ may have wireswound around the pad 51A. The wire paths may be substantially normal tothe tool axis, going in the front, back, and on sides of the pad 51A.With the configuration shown in FIG. 4A, the measurements would bezx-measurements.

FIG. 4B is a schematic of a transmitter for another embodiment accordingto the present disclosure. Transmitters 101A″, 101B″ may be electricdipoles normal to the tool axis.

FIG. 5 is a flow chart of one method 500 for estimating a fluidsaturation according to one embodiment of the present disclosure. Instep 510, electromagnetic tool 51A may be conveyed in the borehole 12.In step 520, signals at a plurality of frequencies may be transmittedfrom transmitters 101A, 101B into the earth formation. In someembodiments, at least one of the plurality of frequencies may be at afrequency at or above 500 MHz. In step 530, receivers 103A, 103B maygenerate an output indicative the complex dielectric permittivity of theearth formation 13. In step 540, a rate of change of the imaginary partof the complex dielectric permittivity relative to the real part of thecomplex dielectric permittivity may be estimated using the generatedoutput. In step 550, a fluid saturation may be estimated using theestimated rate of change.

The selection of the plurality of frequencies may include frequencies ator near the high frequency limit of the real part of dielectricpermittivity for the particular polarization type of the earthformation. Several basic polarization types depending on colloidstructure of oil, water contact with the containing porous medium, andwater-oil contact in the containing medium can be identified. Thephysical bases in these cases correspond to migration polarization (theMaxwell-Wagner polarization) at the contacts between colloid particlesin oil, polarization of the double layer and bulk charge at the contactsbetween water and the rock matrix of the containing porous medium, etc.Each polarization type may be identified with a specific structural unitof the medium and cataloged in the dielectric spectra. Determination ofthe particle type in the colloid solution in the porous medium may bethen reduced to the problem of identification of the cataloged and themeasured spectra. It should be noted that polarization types may bereduced to the following three basic polarization types.

Havriliak-Negami relaxation (its specific cases are the Cole-Davidson,Debye, and Cole-Cole relaxations) characterized by frequency dependenceof the complex value of dielectric permittivity may be expressed as:∈*=∈_(∞)+(∈_(s)−∈_(∞))[1+(iωτ)^(1-α)]^(−β)  (8)∈*=∈′−i∈″  (9)m-th power law relaxation∈*=A·(iω)^(−m)  (10)and Maxwell-Wagner relaxation∈*=∈_(∞)+4πσ/iω+(∈_(s)−∈_(∞))/(1+iωτ)  (11).

FIG. 6 shows a set of complex permittivity curves depicting spectralimages of dielectric permittivity for different relaxations related topolarization types and associated with colloidal structure of fluidwithin a saturated porous medium. The Maxwell-Wagner relaxationdetermines the phase of colloid particles, the m-th power law relaxationcorresponds to laminated or disk-shaped micellar colloid particles, theDebye relaxation corresponds to crystalline solid colloid particles, theCole-Davidson relaxation corresponds to local crystalline structuring incolloid particles, etc. Having compiled the catalog of polarizationtypes, the structure of water-oil mixture and colloid oil contents inthe saturated porous medium may be identified. The structure of thewater-oil mixture may be indicative of permeability of the earthformation. For example, water-saturated sandstones and dolomites mayhave a polarization type that demonstrates Cole-Cole relaxation. Thepolarization type may be identified using electromagnetic logging of theborehole. In the kHz range, fresh water-saturated porous media may becharacterized by high dielectric permittivity (up to 10³-10⁴) atcharacteristic relaxation frequencies.

FIG. 7 shows a chart with a set of curves representing the frequencydependence complex dielectric permittivity of sandstone saturated with(i) water and (ii) a mixture of water and transformer oil. Curve 710indicates the real part of dielectric permittivity for sandstonesaturated with water. Curve 720 indicates the imaginary part ofdielectric permittivity for sandstone saturated with water. Curve 730indicates the real part of dielectric permittivity for sandstonesaturated with a mixture of water and transformer oil. Curve 740indicates the imaginary part of dielectric permittivity for sandstonesaturated with a mixture of water and transformer oil.

In the case when both water and oil are present in the earth formation13, step 550 may include finding a bulk fraction of water if there is nooil and a bulk fraction of water in the presence of oil. The differencebetween these two values may yield the bulk fraction of oil. The methodof finding water saturation and oil saturation is illustrated belowusing the cases of the Cole-Cole relaxation and the Havriliak-Negamirelaxation. The estimate of the dielectric permittivity may assume anatural porous medium, such as sandstone or dolomite, that is saturatedwith water and assuming that the dielectric polarization type of thissystem was established via borehole measurements.

FIG. 8 shows an exemplary set of curves for sandstone expressing complexdielectric permittivity over a range of frequencies. In water-saturatedsandstones, the Cole-Cole relaxation is typically observed as shown inas curve 810. The dielectric spectra may be characterized by asymmetrical polarization curve of the Cole-Cole type (the curve on theplane ∈″,∈′) as expressed by the formula:

$\begin{matrix}{ɛ = {ɛ_{\infty} + \frac{ɛ_{s} - ɛ_{\infty}}{1 + \left( {i\;\omega\;\tau} \right)^{1 - \alpha}}}} & (12)\end{matrix}$where ∈ is complex dielectric permittivity, ∈_(∞) the asymptotic valueof the real part of this polarization type at high frequencies, ∈_(s) isthe static value of dielectric permittivity, τ is relaxation time, α isa parameter ranging from 0 to 1, which characterizes the polarizationangle. It has been established experimentally that natural media likesandstones or dolomites demonstrate that the asymptotic value of thereal part of dielectric permittivity depends only on water saturation ofthe pore space and does not depend on the saline concentration insaturating water and rock type. In other words, ∈_(∞) is a universalfunction of water saturation of the porous rock. Frequency dependenciesof the real and imaginary parts of dielectric permittivity forwater-saturated formations (real part, imaginary part) shown in curve810 appear to be symmetrical with respect to the maximum of theimaginary part of dielectric permittivity. Curve 820 may represent theHavriliak-Negami relaxation is observed when oil is present. When oil ispresent, the symmetry in the high frequency domain may no longer remainas shown with curve 820. Experimental data points 820 a-f (at 10 kHz, 50kHz, 100 kHz, 500 kHz, 1.5 MHz, and 55 MHz, respectively) confirm theclose relationship between practice and the Cole-Cole relaxation incurve 810.

The distortion angles 830, 840 between each of the spectral dielectriccurves 810, 820 and the x-axis may depend on the bulk fraction of theoil present in the formation. Distortion may be obtained from thespectral dependences of the dielectric constant (the real and imaginaryparts) in the low frequency domain. Knowing the distortion angle, thebulk fraction of the oil present in the formation may be determined.

For Cole-Cole relaxation it is known that:∈″_(max)=(∈_(s)−∈_(∞))·tan [(1−α)π/4]/2  (13)where ∈″_(max) is the maximal loss factor, ∈_(s) is the static value ofthe real part of dielectric, ∈_(∞) is its high-frequency limit, and α isthe polarization parameter. It is also known, for Cole-Cole relaxation,that the following relationship is true.∈_(s)=2∈′_(max)−∈_(∞)  (14)

Using eqns. (13) and (14), it follows that:∈_(∞)=∈′_(max)−∈″_(max)/tan [(1−α)π/4]/2=∈_(∞)(K)  (15)where ∈_(∞)(K) is a given universal function of water saturation (waterfraction in percent). The universal curve ∈_(∞)(K %) may be obtained vialaboratory experiments, and ∈″_(max), ∈′_(max), α a may be obtained viainductive logging.

If both water and oil are present in the porous space, the polarizationcurve follows the Havriliak-Negami formula as shown as curve 820 in FIG.8 and expressed as follows.

$\begin{matrix}{ɛ^{*} = {ɛ_{\infty} + \frac{ɛ_{s} - ɛ_{\infty}}{\left\lbrack {1 + \left( {i\;\omega\;\tau} \right)^{1 - \alpha}} \right\rbrack^{\beta}}}} & (16)\end{matrix}$where there are two polarization parameters: α and β.

The presence of the β term may result in the high-frequency limit of thereal part of complex dielectric permittivity depending in part on oilcontents. For small distortions of the left angle in curve 820,expansion with respect to the power indices β−1 may be performed withthe accuracy of quadratic terms and has the following form:∈_(∞)=∈_(∞,0)+∈_(s)(β−1)/2+ . . .   (17)

The first term of this expansion may describe angle distortionscorrectly up to the values of β=0.7. In the case of a further decreaseof β, the quadratic terms are to be taken into account. When oil ispresent in the porous space, water saturation may be calculated for theno-oil case K_(water) using the right hand angle. Polarizationparameters ∈″_(max) and ∈′_(max) may be found via the right hand angle.Based on the polarization curve for the oil case, β=0.7 may becalculated using the eqn (13). Using the following dependence:∈_(∞,0)+∈_(s)(β−1)/2=∈_(∞)(K _(o-w))  (18)the water fraction in the presence of oil K_(o-w) may be calculated. Oilcontent K_(oil) may then found via the following formula:K _(oil) =K _(water) −K _(o-w)  (19)

This example using the Cole-Cole relaxation case is illustrative andexemplary only, as other polarization types may be used, including, butnot limited to, Debye relaxation, Cole-Davidson relaxation, m-th powerlaw relaxation, and Maxwell-Wagner relaxation.

Comparative analysis of the spectral dependence of thewater-and-oil-saturated porous media and the spectra of water-oilemulsions shows that, in the porous medium, oil seems to have the formof water-oil emulsion, i.e. there are droplets of oil in water insidethe pores. The physics of polarization losses in the MHz range is asfollows.

The Debye polarization (a special case of the Cole-Cole polarizationwith a single relaxation time of the corresponding distributionfunction) characterizes polarization in the system of independentoscillators in the external electric field. Asymmetry of the Debyepolarization curve is related to the Cole-Davidson polarization curve.The physics behind this deformation of the polarization curve may beemerging non-linear interaction in the system of independent oscillators(polarizing dipoles in the external electric field). The analysis of theexperimental spectral data for dielectric permittivity of the porousmedia saturated with water-oil mixture and for water-oil emulsionsappears to indicate that, in the porous media, oil takes the form ofdroplets in water, i.e. a system similar to that of water-oil emulsionsin the porous media.

More generally, the dielectric polarization spectrum for a porousreservoir formed by rock (sandstones, carbonates, etc.) saturated withoil and water mixture may have the form of Havriliak-Negami spectraldependence:

$\begin{matrix}{ɛ^{*} = {ɛ_{\infty} + {\frac{ɛ_{0} + ɛ_{\infty}}{\left\lbrack {1 + \left( {i\;\omega\;\tau} \right)^{1 - a}} \right\rbrack^{b}}.}}} & (20)\end{matrix}$

In this formula, ∈*=∈′+i∈″ is the complex value of dielectricpermittivity; ∈_(∞) is an asymptotic value (ω=∞) of dielectricpermittivity; ∈₀ is an asymptotic value (ω=0) of dielectricpermittivity; τ is relaxation time; ω is the frequency ofelectromagnetic excitation of the medium; and a and b are parameterscharacterizing the porous medium (0≦a<1; 1≧b>0).

The Cole-Cole diagram shows that the ∈″(∈′) dependence has the form ofan arc with an asymmetrically sloped left side (FIG. 8). Assuming thatthe curve shows the spectral characteristics of the Havriliak-Negamirelaxation curve, experimental results show that

$\begin{matrix}{v = \frac{2\; ɛ_{\max}^{''}}{ɛ_{0} - ɛ_{\infty}}} & (21)\end{matrix}$is invariant for the water-saturated porous reservoir with differentdegrees of oil saturation. Indeed, for sandstone saturated withdistilled water, we have ∈₀−∈_(∞)=154.5, ∈″_(max)=61, ν=0.79. For thesame sample, but with 54% (of pore volume) transformer oil saturation,we have: ∈₀−∈_(∞)=124.5, ∈″_(max)=48, ν=0.77.

Based on the measured characteristics ∈_(∞), ∈₀, τ, α, β, we plot thedielectric spectrum ∈″(∈′) and calculate ν. This value does not seem tochange with changes in proportions of oil and water saturation for thesame porous medium. Thus, ν characterizes a reservoir saturated withwater.

The following dependence holds true for the Havriliak-Negami relaxation:

$\begin{matrix}{{\frac{2\; ɛ_{\max}^{''}}{ɛ_{0} - ɛ_{\infty}} = {{{tg}\left( {\frac{1 - \alpha}{4}\pi} \right)} \cdot {\chi(\beta)}}},{{\chi(\beta)} = {2\;{{{Sin}\left\lbrack \frac{\beta\;\pi}{2\left( {1 + \beta} \right)} \right\rbrack}^{1 + \beta}.}}}} & (22)\end{matrix}$

The parameter β may be considered as the parameter characterizing thedegree to which the porous formation is saturated with oil, in thepresence of water. Its value determines the value of the parameterα=α(β). Thus, solving equations (22) for α=α(β), we can find dependence

$\begin{matrix}{{\alpha(\beta)} = {1 - {\frac{4}{\pi}{{arc}{tg}}{\frac{v}{\chi(\beta)}.}}}} & (23)\end{matrix}$

The parameter ν depends on rock porosity and lithology, as well as onthe electrochemical characteristics of the contact between water androck, for the porous medium. FIG. 9A shows the graph of α=α(β) for twovalues of ν. FIG. 9B shows the graph of α_(*)=α_(*)(β_(*)). Oilsaturation β varies from 1 to β_(*). The point β=1 corresponds to thestate of the medium with no oil in the reservoir. The point β=β_(*)corresponds to the state of the reservoir saturated with oil only (nowater). The current value of β corresponds to the current value of αfound for any ν, as calculated above. The parameter α may vary from 0 toα_(*), as is apparent from the graph. The point α=0 corresponds to thestate of the medium with no water in the water-oil mixture. The pointα=α_(*) corresponds to the state of the reservoir with no oil (wateronly).

The parameters α_(*) and β_(*) are related as follows:

$\begin{matrix}{\alpha_{*} = {1 - {\frac{4}{\pi}{{{{arc}{tg}}\left\lbrack {\chi\left( \beta_{*} \right)} \right\rbrack}.}}}} & (24)\end{matrix}$

Oil saturation corresponds to 1−β with a possibility of oil saturationreaching 1−β_(*). Water saturation corresponds to α with a possibilityof water saturation reaching α_(*). Therefore, the percentage of oil maybe described with the following formula:

$\begin{matrix}{{{K_{oil}(\%)} = {{\frac{1 - \beta}{1 - \beta_{*}} \cdot 100}\%}}{{K_{oil}(\%)} = {{\frac{\alpha(\beta)}{\alpha_{*}\left( \beta_{*} \right)} \cdot 100}\%}}} & (25)\end{matrix}$and the percentage of water may be described with the following formula:

$\begin{matrix}{{K_{water}(\%)} = {{\frac{\alpha}{\alpha_{*}} \cdot 100}{\%.}}} & (26)\end{matrix}$

Formation porosity is water saturation at β_(*)=1 and may be estimatedbased on α=α_(*). Under the lab conditions, for example, for a varietyof lithologies (water-saturated sandstones, carbonates, dolomites,etc.), dependence of α_(*) on water saturation or on pore volume K_(p),may be measured (K_(p)=α_(*)·100%). A table may be created relating thedependence α_(*)=α_(*)(K_(p)). Advantageously, it is possible toestimate α_(*), ν and β_(*)=1 under borehole conditions, via thepolarization characteristics ∈_(∞), ∈₀, τ, α, β.

Examples of calculated values may include, for sandstones:2∈″_(max)/Δ∈=0.79, α_(*)=0.14, K _(p)=α_(*)·100%=14 percentwhich approximates an experimentally derived value of 14.1 percent, andfor dolomites:2∈″_(max)/Δ∈=0.74, α_(*)=0.18, K _(p)=α_(*)·100%=18 percentwhich approximates an experimentally derived value of 16.9 percent.

Any current value of β may help compare oil fraction (percent) in awater- and oil-saturated porous reservoir using equations (25), andporosity may be determined from the tabulated curve.

Thus, we have the borehole method of finding changes in oil fraction(percent) in the water-oil reservoir and its porosity.

FIG. 10 is a flow chart of one method 1000 for estimating changes in oilfraction in the water-oil reservoir and its porosity according to oneembodiment of the present disclosure. In step 1010, electromagnetic tool51A may be conveyed in the borehole 12. In step 1020, signals at aplurality of frequencies may be transmitted from transmitters 101A, 101Binto the earth formation. In some embodiments, at least one of theplurality of frequencies may be at a frequency at or above 500 MHz. Instep 1030, receivers 103A, 103B may generate an output indicative thecomplex dielectric permittivity of the earth formation 13. In step 1040,one or more of an oil saturation parameter and a water saturationparameter for a complete saturation point (β_(*) and α_(*),respectively) may be estimated using the generated output. In step 1050,at least one of the formation porosity, the water fraction, and the oilfraction may be estimated using the estimated parameters, β_(*) andα_(*). In some embodiments, the parameters β_(*) and α_(*) may beestimated using an output in the form of a Havriliak-Negami polarizationcurve, such as with sandstone. One of skill in the art, with the benefitof the teachings in this disclosure, would understand that differentpolarization curves may be used depending on lithology, such as, but notlimited to, the Cole-Davidson polarization curve.

As described herein, the method in accordance with the presentlydisclosed embodiment of the disclosure involves several computationalsteps. As would be apparent by persons of ordinary skill, these stepsmay be performed by computational means such as a computer, or may beperformed manually by an analyst, or by some combination thereof. As anexample, where the disclosed embodiment calls for selection of measuredvalues having certain characteristics, it would be apparent to those ofordinary skill in the art that such comparison could be performed basedupon a subjective assessment by an analyst or by computationalassessment by a computer system properly programmed to perform such afunction. To the extent that the present disclosure is implementedutilizing computer equipment to perform one or more functions, it isbelieved that programming computer equipment to perform these stepswould be a matter of routine engineering to persons of ordinary skill inthe art having the benefit of the present disclosure.

Implicit in the processing of the acquired data is the use of a computerprogram implemented on a suitable computational platform (dedicated orgeneral purpose) and embodied in a suitable machine readable medium thatenables the processor to perform the control and processing. The term“processor” as used in the present disclosure is intended to encompasssuch devices as microcontrollers, microprocessors, field-programmablegate arrays (FPGAs) and the storage medium may include ROM, RAM, EPROM,EPROM, solid-state disk, optical media, magnetic media and other mediaand/or storage mechanisms as may be deemed appropriate. As discussedabove, processing and control functions may be performed downhole, atthe surface, or in both locations.

From the foregoing disclosure, it should be apparent that a method andapparatus for evaluating an earth formation has been disclosed involvingthe measurement of electrical characteristics including formationdielectric permittivity and involving measurements taken at a pluralityof measurement frequencies.

Although a specific embodiment of the disclosure as well as possiblevariants and alternatives thereof have been described and/or suggestedherein, it is to be understood that the present disclosure is intendedto teach, suggest, and illustrate various features and aspects of thedisclosure, but is not intended to be limiting with respect to the scopeof the disclosure, as defined exclusively in and by the claims, whichfollow.

While the foregoing disclosure is directed to the specific embodimentsof the disclosure, various modifications will be apparent to thoseskilled in the art. It is intended that all such variations within thescope of the appended claims be embraced by the foregoing disclosure.

What is claimed is:
 1. A method of evaluating an earth formationincluding a porous medium, the method comprising: making a plurality ofestimates of complex permittivity based on measurements at at least onereceiver on an electromagnetic tool responsive to electromagneticsignals transmitted at a plurality of frequencies in a boreholepenetrating the earth formation from a transmitter in the borehole; andestimating a parameter of interest of the earth formation using theplurality of estimates by using a spectral dielectric curve constant (v)for the porous medium, the spectral dielectric curve constant (v)invariant with respect to water-oil ratio in the porous medium anddetermined by an estimated rate of change between estimates of theplurality of estimates of complex permittivity of an imaginary partrelative to a real part of each estimate, comprising: generating aspectral dielectric curve by mapping the real part with respect to theimaginary part for each estimate of the plurality of estimates; using atleast one processor to perform at least one of: i) storing the parameterin a computer memory; ii) transmitting the parameter uphole; or iii)displaying of the parameter to an operating engineer.
 2. The method ofclaim 1, wherein the spectral dielectric curve constant (v) iscalculated using a maximum imaginary value for the spectral dielectriccurve and asymptotic values (∈=∈₀, ∈=∈_(∞)) for the real parts of eachestimate of the spectral dielectric curve.
 3. The method of claim 2,further comprising: inverting the measurements to generate the imaginarypart and the real part of each estimate of complex permittivity at theplurality of frequencies; and estimating the rate of change of theimaginary part relative to the real part from one of the estimates toanother of the estimates.
 4. The method of claim 1, comprising using amodel correlating the plurality of estimates to a Havrilyaka-Negamirelaxation curve to determine a value for at least one polarizationparameter associated with the curve.
 5. The method of claim 4, whereinthe at least one polarization parameter comprises β, wherein β relatesto a degree of saturation of the porous medium with oil in the presenceof water, and the value for β comprises β₀.
 6. The method of claim 5,comprising determining the value for β using an angle of intersection(φ) of two curves Im∈=Φ(Re∈) and Im ∈=0 at the point where ∈=∈₀ and asecond angle of intersection (ψ) of the two curves Im∈=Φ(Re∈) and Im ∈=0at the point where ∈=∈_(∞).
 7. The method of claim 5, comprisingmodeling an other polarization parameter α as a function of the spectraldielectric curve constant (v) and the value for β, wherein α relates toa degree of saturation of the porous medium with water, and wherein αrelates to a polarization angle.
 8. The method of claim 7, comprisingdetermining a value α_(*) of the other polarization parametercorresponding to a second value for β correlated with an oil-free stateof the porous medium.
 9. The method of claim 8, comprising using thevalues α and α_(*) to estimate a bulk water fraction of the porousmedium.
 10. The method of claim 8, comprising using the values α andα_(*) to estimate a bulk oil fraction of the porous medium.
 11. Themethod of claim 5, comprising: determining a value β_(*) using arelationship defining β_(*) as a function of α, wherein β_(*)corresponds to a value for α correlated with a water-free state of theporous medium; and using the values β and β_(*) to estimate a bulk oilfraction of the porous medium.
 12. The method of claim 5, comprising:determining a value β_(*) using a relationship defining β_(*) as afunction of α, wherein β_(*) corresponds to a value for α correlatedwith a water-free state of the porous medium; and using the value β_(*)to estimate a porosity of the porous medium.
 13. The method of claim 1,further comprising using the electromagnetic tool for making themeasurements at the plurality of frequencies.
 14. The method of claim 1,wherein the electromagnetic tool uses electrical induction.
 15. Themethod of claim 1, further comprising: conveying the electromagnetictool in the borehole.
 16. An apparatus for evaluating an earthformation, the apparatus comprising: a carrier configured to be conveyedin a borehole penetrating the earth formation; a electromagnetic tooldisposed on the carrier and configured to make a plurality of estimatesof complex permittivity based on measurements at at least one receiveron the tool responsive to electromagnetic signals transmitted at aplurality of frequencies in a borehole penetrating the earth formationfrom a transmitter in the borehole; and a processor configured toestimate a parameter of interest of the earth formation by using aspectral dielectric curve constant (v) for the porous medium, thespectral dielectric curve constant (v) invariant with respect towater-oil ratio in the porous medium and determined by an estimated rateof change between estimates of the plurality of estimates of complexpermittivity of an imaginary part relative to a real part of eachestimate, comprising: generating a spectral dielectric curve by mappingthe real part with respect to the imaginary part for each estimate ofthe plurality of estimates; using at least one processor to perform atleast one of: i) storing the parameter in a computer memory; ii)transmitting the parameter uphole; or iii) displaying of the parameterto an operating engineer.